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Theorem mulgsubdir 15966
Description: Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgsubdir.b  |-  B  =  ( Base `  G
)
mulgsubdir.t  |-  .x.  =  (.g
`  G )
mulgsubdir.d  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
mulgsubdir  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  -  N )  .x.  X )  =  ( ( M  .x.  X
)  .-  ( N  .x.  X ) ) )

Proof of Theorem mulgsubdir
StepHypRef Expression
1 znegcl 10887 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2 mulgsubdir.b . . . 4  |-  B  =  ( Base `  G
)
3 mulgsubdir.t . . . 4  |-  .x.  =  (.g
`  G )
4 eqid 2460 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
52, 3, 4mulgdir 15960 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  -u N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) ) )
61, 5syl3anr2 1276 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) ) )
7 simpr1 997 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  M  e.  ZZ )
87zcnd 10956 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  M  e.  CC )
9 simpr2 998 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  N  e.  ZZ )
109zcnd 10956 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  N  e.  CC )
118, 10negsubd 9925 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( M  +  -u N )  =  ( M  -  N
) )
1211oveq1d 6290 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  -  N )  .x.  X
) )
13 eqid 2460 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
142, 3, 13mulgneg 15953 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( ( invg `  G ) `
 ( N  .x.  X ) ) )
15143adant3r1 1200 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( -u N  .x.  X )  =  ( ( invg `  G ) `  ( N  .x.  X ) ) )
1615oveq2d 6291 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) )  =  ( ( M  .x.  X ) ( +g  `  G ) ( ( invg `  G
) `  ( N  .x.  X ) ) ) )
172, 3mulgcl 15952 . . . . 5  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
18173adant3r2 1201 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( M  .x.  X )  e.  B
)
192, 3mulgcl 15952 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
20193adant3r1 1200 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
21 mulgsubdir.d . . . . 5  |-  .-  =  ( -g `  G )
222, 4, 13, 21grpsubval 15887 . . . 4  |-  ( ( ( M  .x.  X
)  e.  B  /\  ( N  .x.  X )  e.  B )  -> 
( ( M  .x.  X )  .-  ( N  .x.  X ) )  =  ( ( M 
.x.  X ) ( +g  `  G ) ( ( invg `  G ) `  ( N  .x.  X ) ) ) )
2318, 20, 22syl2anc 661 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X )  .-  ( N  .x.  X ) )  =  ( ( M  .x.  X ) ( +g  `  G
) ( ( invg `  G ) `
 ( N  .x.  X ) ) ) )
2416, 23eqtr4d 2504 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) )  =  ( ( M  .x.  X )  .-  ( N  .x.  X ) ) )
256, 12, 243eqtr3d 2509 1  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  -  N )  .x.  X )  =  ( ( M  .x.  X
)  .-  ( N  .x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275    + caddc 9484    - cmin 9794   -ucneg 9795   ZZcz 10853   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   invgcminusg 15717   -gcsg 15719  .gcmg 15720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-seq 12064  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854
This theorem is referenced by:  odmod  16359  odcong  16362  gexdvds  16393  archiabllem1a  27383
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